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Math Expert V
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If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Question Stats: 20% (02:33) correct 80% (01:55) wrong based on 222 sessions

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If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

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If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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2
Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

Interesting question

Here the question says rain on atleast three days CONSECUTIVELY
i.e. Exactly three days in row Days123 or days 234 or days 345 or days 1235 or days 1345
i.e. Exactly Four days Days1234 or days 2345
or exactly 5 days days 12345
Probability in each case = (1/2)^5

Required Probability = 8*(1/32) = 1/4

A VARIATION AND A NEW QUESTION
Had the question been Probability of Rain on Atleast three days out of 5

If it rains on exactly 3 days out of 5 then those 3 days can be selected in 5C3 ways and therefore Probability = $$5C3*(0.5)^3*(0.5)^2 = 5C3*(1/32)$$

If it rains on exactly 4 days out of 5 then those 3 days can be selected in 5C4 ways and therefore Probability = $$5C4*(0.5)^4*(0.5)^1 = 5C4*(1/32)$$

If it rains on exactly 5 days out of 5 then those 3 days can be selected in 5C5 ways and therefore Probability = $$5C5*(0.5)^5 = 5C3*(1/32)$$

Total Probability of raining atleast 3 out of 5 days = (1/32)*(5C3+5C4+5C5) = 16/32 = 1/2
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Math Expert V
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

Important not to miss ' in a row'..
At least three days means three cases...
1) 3 days in a row..
So take these three days as ONE R and other two as N
So NNR..
You can place R in 3C1=3 ways

2) 4 days.....4 days in a row OR 3 days in a row and 4th day of rain separated by a non rainy day
So take these three days as ONE R and non-rainy day as N and 4th day as r
So rNR........ rNR, rRN, RNr,rRN
4 ways

3) 5 days in a row
All 5 are one so one way

Total- 3+4+1=8
All possibilities=2*2*2*2*2=32

Probability=8/32=1/4

gmatnovice1122, what is wrong with your way..
Firstly you are complicating a simple way..
We use complement when we went to ease our calculations as normal scenario will have more cases
If talking of complement..
At least three days in a row means MAX two days in a row..
1) two days..
When there are 4 days of rains....RRNRR
When there are three days....RNNRR, NRNRR, ....many ways
When there are 2 days....RRNNN, NRRNN..and so on
2) one day..
4days is not possible
3 days..RNRNR
2 days ...NNRNR, and so on
3) none of the day
NNNNN... One way

So as you see it is more complicated, stick to normal
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

$$5C_3(\frac{1}{2})^3(\frac{1}{2})^2$$

$$= \frac{5}{16}$$ , Answer must be (D)
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If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Solution

Given:
• Probability of rain on any given day is 50%

To find:
• The probability that it will rain on at least three days in a row during a five-day period

Approach and Working:
Assume that R represents rain and N represents no rain
As probability of rain is 50%, each of R and N have individual probability ½.

Now, at least 3 rainy days in a row can happen when
• Exact 3 rain and 2 no rain: RRRNN or NRRRN or NNRRR or RRRNR or RNRRR
• Exact 4 rain and 1 no rain: RRRRN or NRRRR
• Exact 5 rain: RRRRR

Hence, the probability = (5 + 2 + 1) x $$(\frac{1}{2})^5$$ = $$\frac{8}{32}$$ = $$\frac{1}{4}$$

Hence, the correct answer is option B.

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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Abhishek009 wrote:
$$5C_3(\frac{1}{2})^3(\frac{1}{2})^2$$

$$^5C_3$$ denotes any three days in the 5-day period, but the question asks for only those instances where it rained for at least 3 days in a row. You may have missed that.
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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2
Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

Say , symbols R=Rain , N=No rain
Given $$P(R)=P(N)=\frac{1}{2}$$

P(at least 3 Rainy days in a row)=P(3 rainy days in a row)+P(4 rainy days in a row)+P(5 rainy days in a row)

1) P(3 rainy days in a row)=$$5*(\frac{1}{2})^5$$
RRRNR
RRRNN
NNRRR
RNRRR
NRRRN

2) P( 4 rainy days in a row)=$$2*(\frac{1}{2})^5$$
RRRRN
NRRRR

3) P( 5 rainy days in a row)=$$(\frac{(1}{2)})^5$$
RRRRR

So, required probability=$$5*(\frac{1}{2})^5$$+$$2*(\frac{1}{2})^5$$+$$(\frac{1}{2})^5$$=$$8*(\frac{1}{2})^5$$=$$8*\frac{1}{32}$$=$$\frac{1}{4}$$

Ans. (B)
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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'Coin Flip' analogy

5 flips; what is the Probability of getting atleast three tails in a row?

Probability of 1 event= 1/2^5

Favorable events (by manual counting):

1) Three tails in a row: 5
TTTHH + TTTHT
HTTTH
HHTTT + THTTT

2) Four tails in a row: 2
TTTTH
HTTTT

3) Five tails in a row: 1
TTTTT

Ans: 8/2^5= 8/32= 1/4
Choice B

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If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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EgmatQuantExpert wrote:

Solution

Given:
• Probability of rain on any given day is 50%

To find:
• The probability that it will rain on at least three days in a row during a five-day period

Approach and Working:
Assume that R represents rain and N represents no rain
As probability of rain is 50%, each of R and N have individual probability ½.

Now, at least 3 rainy days in a row can happen when
• Exact 3 rain and 2 no rain: RRRNN or NRRRN or NNRRR or RRRNR or RNRRR
• Exact 4 rain and 1 no rain: RRRRN or NRRRR
• Exact 5 rain: RRRRR

Hence, the probability = (5 + 2 + 1) x $$(\frac{1}{2})^5$$ = $$\frac{8}{32}$$ = $$\frac{1}{4}$$

Hence, the correct answer is option B.

Hi EgmatQuantExpert

For below scenario, why do we also need to count the last two cases? That's when there are 4 rainy days.

Exact 3 rain and 2 no rain: RRRNN or NRRRN or NNRRR or RRRNR or RNRRR
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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1
1
Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

Let R = rain and N = no rain. To have at least three days in a row during a five-day period, we could have:

RRRNN, NRRRN, NNRRR, RRRRN, NRRRR, RNRRR, RRRNR and RRRRR.

Notice that the probability of R is ½ and the probability of N is also ½. Thus, the probability of any of the scenarios above is ½ x ½ x ½ x ½ x ½ = (½)^5 = 1/32. Since we have a total of 6 scenarios, the overall probability is 8 x 1/32 = 1/4.

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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Can anyone please explain how do you solve this using the complement "At Least" rule?
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period? Can we solve using the opposite or negation role in which case N - 3 = 2.

P(it will rain on at least three days in a row) = 1 - P(it will rain on at least two or fewer days in a row).
P(it will rain on at least two or fewer days in a row) = Raining on any one week day = 5/32 (RNNNN,NRNNN, NNRNN,NNNRN,NNNNR)
Raining on any two consecutive days in a row = 4 / 32 (RRNNN, NRRNN, NNRRN, NNNRR)

4/32 + 5/32 = 9/32
1 - 9/32 = 23/32 (It is not one of the answer)!
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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ScottTargetTestPrep wrote:
Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

Let R = rain and N = no rain. To have at least three days in a row during a five-day period, we could have:

RRRNN, NRRRN, NNRRR, RRRRN, NRRRR, RNRRR, RRRNR and RRRRR.

Notice that the probability of R is ½ and the probability of N is also ½. Thus, the probability of any of the scenarios above is ½ x ½ x ½ x ½ x ½ = (½)^5 = 1/32. Since we have a total of 6 scenarios, the overall probability is 8 x 1/32 = 1/4.

Hi ScottTargetTestPrep

Can anyone please explain how do you solve this using the complement "At Least" rule?
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period? Can we solve using the opposite or negation role in which case N - 3 = 2.

P(it will rain on at least three days in a row) = 1 - P(it will rain on at least two or fewer days in a row).
P(it will rain on at least two or fewer days in a row) = Raining on any one week day = 5/32 (RNNNN,NRNNN, NNRNN,NNNRN,NNNNR)
Raining on any two consecutive days in a row = 4 / 32 (RRNNN, NRRNN, NNRRN, NNNRR)

4/32 + 5/32 = 9/32
1 - 9/32 = 23/32 (It is not one of the answer)!
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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It was a quite confusing one took me around 10 mins to understand the solution.
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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parthos wrote:
It was a quite confusing one took me around 10 mins to understand the solution.

parthos
I understand the solution but why cannot we solve it with the complement.
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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alirehman4657 wrote:
parthos wrote:
It was a quite confusing one took me around 10 mins to understand the solution.

parthos
I understand the solution but why cannot we solve it with the complement.

I believe the approach used by you to find solution is incorrect in this type of question and as I told you in the chat I am not expert in probability hence please contact some seniors for this.

Thanks !!
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If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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chetan2u wrote:
Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

Important not to miss ' in a row'..
At least three days means three cases...
1) 3 days in a row..
So take these three days as ONE R and other two as N
So NNR..
You can place R in 3C1=3 ways

2) 4 days.....4 days in a row OR 3 days in a row and 4th day of rain separated by a non rainy day
So take these three days as ONE R and non-rainy day as N and 4th day as r
So rNR........ rNR, rRN, RNr,rRN
4 ways

3) 5 days in a row
All 5 are one so one way

Total- 3+4+1=8
All possibilities=2*2*2*2*2=32

Probability=8/32=1/4

what is wrong with your way..
Firstly you are complicating a simple way..
We use complement when we went to ease our calculations as normal scenario will have more cases
If talking of complement..
At least three days in a row means MAX two days in a row..(IS THE COMPLEMENT OF AT LEAST THREE DAYS IN A ROW = 1 - P(MAX TWO DAYS)???)
1) two days..
When there are 4 days of rains....RRNRR
When there are three days....RNNRR, NRNRR, ....many ways
When there are 2 days....RRNNN, NRRNN..and so on
2) one day..
4days is not possible
3 days..RNRNR
2 days ...NNRNR, and so on
3) none of the day
NNNNN... One way

So as you see it is more complicated, stick to normal

chetan2u
GMATinsight
Thanks for your respond. I appreciate the feedback.
I was confused because when reviewing probabilities, you are thought that when you have the words "AT LEAST" to automatically use the complement. As a result, that is the method I followed.
Also, wouldn't the complement of P(Rain on AT LEAST three days in a row) be 1 - P(Rain on AT LEAST two or fewer days in a row. and NOT as you stated above..so it would be RRNNN,NRRNN, etc? Is my understanding of complements wrong?

In the past I solved a problem related to selecting cards out of a 52 card deck, that stated P (at least the third draw). The complement of this was used 1 - p(two or fewer draws).

Thanks again for helping clarify this.
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Joined: 02 Aug 2009
Posts: 8335
If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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gmatnovice1122 wrote:
chetan2u wrote:
Bunuel wrote:
If the probability of rain on any given day is 50%, what is the probability that it will rain on at least three days in a row during a five-day period?

A. 3/32
B. 1/4
C. 9/32
D. 5/16
E. 1/2

Important not to miss ' in a row'..
At least three days means three cases...
1) 3 days in a row..
So take these three days as ONE R and other two as N
So NNR..
You can place R in 3C1=3 ways

2) 4 days.....4 days in a row OR 3 days in a row and 4th day of rain separated by a non rainy day
So take these three days as ONE R and non-rainy day as N and 4th day as r
So rNR........ rNR, rRN, RNr,rRN
4 ways

3) 5 days in a row
All 5 are one so one way

Total- 3+4+1=8
All possibilities=2*2*2*2*2=32

Probability=8/32=1/4

what is wrong with your way..
Firstly you are complicating a simple way..
We use complement when we went to ease our calculations as normal scenario will have more cases
If talking of complement..
At least three days in a row means MAX two days in a row..(IS THE COMPLEMENT OF AT LEAST THREE DAYS IN A ROW = 1 - P(MAX TWO DAYS)???)
1) two days..
When there are 4 days of rains....RRNRR
When there are three days....RNNRR, NRNRR, ....many ways
When there are 2 days....RRNNN, NRRNN..and so on
2) one day..
4days is not possible
3 days..RNRNR
2 days ...NNRNR, and so on
3) none of the day
NNNNN... One way

So as you see it is more complicated, stick to normal

chetan2u
GMATinsight
Thanks for your respond. I appreciate the feedback.
I was confused because when reviewing probabilities, you are thought that when you have the words "AT LEAST" to automatically use the complement. As a result, that is the method I followed.
Also, wouldn't the complement of P(Rain on AT LEAST three days in a row) be 1 - P(Rain on AT LEAST two or fewer days in a row. and NOT as you stated above..so it would be RRNNN,NRRNN, etc? Is my understanding of complements wrong?

In the past I solved a problem related to selecting cards out of a 52 card deck, that stated P (at least the third draw). The complement of this was used 1 - p(two or fewer draws).

Thanks again for helping clarify this.

Hi..

At least 3 days means on 3 days, 4 days and 5 days..
So complement will be 1-(no days, 1 days and 2 days)
The moment you do at least 2 days, it itself will be contain at least 3 days or at least 4 days..

So if you do 1-(atleast 2 days or few days) means 1-(probability of rain on 2 days, 3 days , 4 days and 5 days)
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Hi All,

We're told that the probability of rain on any given day is 50%. We're asked for the probability that it will rain on AT LEAST three days in a ROW during a five-day period. This question can be approached in a number of different ways; you might find it easiest to do a little math and 'map out' the various possible ways to get at least 3 days of rain in a row.

Since there are two events (rain or no rain) that have an equal chance of occurring each day, with 5 days in a row, there are (2)(2)(2)(2)(2) = 32 possible outcomes for the 5 days. The number of outcomes that include AT LEAST 3 days of rain IN A ROW are....

3 days:
RRRNN
NRRRN
NNRRR

4 days:
RRRRN
RRRNR
RRNRR
RNRRR
NRRRR

5 days:
RRRRR

8 options out of 32 total = 8/32 = 1/4

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If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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I have tried the complimentary case, i.e. LESS THAN 3 days in a row, and got 24 favorable outcomes as detailed below.

1. 2 days in a row
a) 4 days of rain: RRNRR -> 1 case
b) 3 days of rain: RRNRN RRNNR NRRNR RNRRN RNNRR NRNRR -> 6 cases
c) 2 days of rain: RRNNN NRRNN NNRRN NNNRR -> 4 cases (=4!/3!)

2. 1 day in a row
a) 3 days of rain: RNRNR -> 1 case
b) 2 days of rain: RNRNN RNNRN RNNNR NRNRN NRNNR NNRNR -> 6 cases (=5!/3!2! - 4!/3!)
c) 1 days of rain: RNNNN NRNNN NNRNN NNNRN NNNNR -> 5 cases (=5!/4!)

3. 0 day in a row
a) 0 day of rain: NNNNN -> 1 case

The total outcome is 32. Therefore, the probability for the complimentary case is 24/32.

The probability that we want is 1 - 24/32 = 8/32, which also arrives at the correct answer!

My question is whether this approach is also valid? Or it is a mere coincidence?

Is LESS THAN 3 days in a row the complimentary case for AT LEAST 3 days in a row?
I notice that the permutation works for some cases (as written in the parentheses) and does not for many others. Could you please explain why?

Please help!   Originally posted by varotkorn on 05 Jan 2020, 19:21.
Last edited by varotkorn on 05 Jan 2020, 21:21, edited 1 time in total.
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Re: If the probability of rain on any given day is 50%, what is the probab  [#permalink]

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Hi varotkorn,

When dealing with a Probability question, there are two outcomes that you can calculate: what you WANT to have happen or what you DON'T WANT to have happen. Those two fractions will sum to 1 (which is why you were able to calculate the correct answer by calculating the probability of NOT getting at least 3 days of rain in a row). Both methods are valid - and as a general rule, you should use whichever one is faster given the 'restrictions' in the prompt.

You made a reference to permutation calculations that did not "work" - can you give an example of a calculation that you tried that did not lead to the answer that you were expecting?

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_________________ Re: If the probability of rain on any given day is 50%, what is the probab   [#permalink] 05 Jan 2020, 19:47

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